Chapter 5 Overview
Electric vs Magnetic Comparison
Electric & Magnetic Forces
Magnetic force
Electromagnetic (Lorentz)
force
Magnetic Force on a Current Element
Differential force dFm on a differential current I dl:
Torque
d = moment arm
F = force
T = torque
Magnetic Torque on Current Loop
No forces on arms 2 and 4 ( because I
and B are parallel, or anti-parallel)
Magnetic torque:
Area of Loop
Inclined Loop
For a loop with N turns and whose surface
normal is at angle that relative to B direction:
Biot-Savart Law
Magnetic field induced by
a differential current:
For the entire length:
Magnetic Field due to Current Densities
Example 5-2: Magnetic Field of Linear Conductor
Cont.
Example 5-2: Magnetic
Field of Linear Conductor
Magnetic Field of Long Conductor
Example 5-3: Magnetic Field of a Loop
Magnitude of field due to dl is
dH is in the r–z plane , and therefore it has
components dHr and dHz
z-components of the magnetic fields due to dl and
dl’ add because they are in the same direction,
but their r-components cancel
Hence for element dl:
Cont.
Example 5-3:Magnetic Field of a Loop (cont.)
For the entire loop:
Magnetic Dipole
Because a circular loop exhibits a magnetic field pattern similar to the
electric field of an electric dipole, it is called a magnetic dipole
Forces on Parallel Conductors
Parallel wires attract if their currents are in the same
direction, and repel if currents are in opposite directions
Ampère’s Law
Internal Magnetic Field of Long
Conductor
For r < a
Cont.
External Magnetic Field of Long
Conductor
For r > a
Magnetic Field of Toroid
Applying Ampere’s law over contour C:
Ampere’s law states that the line integral of
H around a closed contour C is equal to the
current traversing the surface bounded by the
contour.
The magnetic field outside the toroid
is zero. Why?
Magnetic Vector Potential A
Electrostatics
Magnetostatics
Magnetic Properties of Materials
Magnetic Hysteresis
Boundary Conditions
Solenoid
Inside the solenoid:
Inductance
Magnetic Flux
Flux Linkage
Inductance
Solenoid
Example 5-7: Inductance of Coaxial Cable
The magnetic field in the region S between
the two conductors is approximately
Total magnetic flux through S:
Inductance per unit length:
Magnetic Energy Density
Magnetic field in the insulating material is
The magnetic energy stored in the
coaxial cable is
Summary